This project of constructing a theory to compute price indices appropriate for the real estate industry was initiated in 1979. At that time a variety of indices were available for financial and consumer markets and these indices had been well established and widely used for decades. Considering the large share of the nation’s wealth represented by real estate, we felt a more accurate price index for real estate was needed. Roger Ibbotson made the initial suggestion that this be done. He also supplied the data set that was used.
A major result of the first paper, A Probabilistic Model for Price Levels in Discontinuous Markets, is the Connectivity Theorem which describes necessary and sufficient conditions on the data for the method of least squares to produce index numbers. This theorem was proved by elementary means utilizing only the arithmetic of matrices and was presented at a Society of Industrial and Applied Mathematics (SIAM) conference in North Carolina in the early 1980s. Charles Johnson who was in attendance pointed out that the theory of M-matrices could be used in its proof. This shortens the proof and was used in the version of the paper presented here.
The other major result of the first paper was adjusting for heteroscedasticity. There is abundant theoretical reason to make this adjustment. But the model was run without this adjustment and the quality of the index numbers produced was compared to the quality of the index numbers produced with the adjustment. A material improvement was noted.
Also in the first paper, a negative correlation between index numbers was noted. This phenomenon was investigated thoroughly in the second paper, The Expected Accuracy of a Price Index for Discontinuous Markets. Even though precise measures of this negative correlation were obtained both theoretically and from an analysis of the data, it remains to be more fully explained why the method of least squares produces correlated estimates whereas the distributions from which the data are obtained are independent and identical.
The second paper also produces estimates on how much data is necessary to produce a given reliability in the index numbers. Many real estate markets have no scarcity of data, but this is not universally true and there are other discontinuous markets for which there is almost always scarcity of data (e.g., art, coins, stamps).
In both papers, an “edited” data set was studied from which presumed outliers had been excluded. It was observed that much better index numbers may be expected if such exclusions are made. Thus a future project is to develop a theory of treating quality variables that could exploit the value of such information. Beyond investigating the “edited” data set, this theory was not developed in either of the two papers. It harkens back to the original work of Andrew Court in 1938.
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